Chapter 12: Monopolistic Competition and Oligopoly 191 CHAPTER 12 MONOPOLISTIC COMPETITION AND OLIGOPOLY EXERCISES 1. Suppose all firms in a monopolistically competitive industry were merged into one large firm. Would that new firm produce as many different brands? Would it produce only a single brand? Explain. Monopolistic competition is defined by product differentiation. Each firm earns economic profit by distinguishing its brand from all other brands. This distinction can arise from underlying differences in the product or from differences in advertising. If these competitors merge into a single firm, the resulting monopolist would not produce as many brands, since too much brand competition is internecine (mutually destructive). However, it is unlikely that only one brand would be produced after the merger. Producing several brands with different prices and characteristics is one method of splitting the market into sets of customers with different price elasticities, which may also stimulate overall demand. 2. Consider two firms facing the demand curve P = 50 - 5Q, where Q = Q 1 + Q 2 . The firms’ cost functions are C 1 (Q 1 ) = 20 + 10Q 1 and C 2 (Q 2 ) = 10 + 12Q 2 . a. Suppose both firms have entered the industry. What is the joint profit-maximizing level of output? How much will each firm produce? How would your answer change if the firms have not yet entered the industry? If both firms enter the market, and they collude, they will face a marginal revenue curve with twice the slope of the demand curve: MR = 50 - 10Q. Setting marginal revenue equal to marginal cost (the marginal cost of Firm 1, since it is lower than that of Firm 2) to determine the profit-maximizing quantity, Q: 50 - 10Q = 10, or Q = 4. Substituting Q = 4 into the demand function to determine price: P = 50 – 5*4 = $30. The question now is how the firms will divide the total output of 4 among themselves. Since the two firms have different cost functions, it will not be optimal for them to split the output evenly between them. The profit maximizing solution is for firm 1 to produce all of the output so that the profit for Firm 1 will be: π 1 = (30)(4) - (20 + (10)(4)) = $60. The profit for Firm 2 will be: π 2 = (30)(0) - (10 + (12)(0)) = -$10. Chapter 12: Monopolistic Competition and Oligopoly Total industry profit will be: π T = π 1 + π 2 = 60 - 10 = $50. If they split the output evenly between them then total profit would be $46 ($20 for firm 1 and $26 for firm 2). If firm 2 preferred to earn a profit of $26 as opposed to $25 then firm 1 could give $1 to firm 2 and it would still have profit of $24, which is higher than the $20 it would earn if they split output. Note that if firm 2 supplied all the output then it would set marginal revenue equal to its marginal cost or 12 and earn a profit of 62.2. In this case, firm 1 would earn a profit of –20, so that total industry profit would be 42.2. If Firm 1 were the only entrant, its profits would be $60 and Firm 2’s would be 0. If Firm 2 were the only entrant, then it would equate marginal revenue with its marginal cost to determine its profit-maximizing quantity: 50 - 10Q 2 = 12, or Q 2 = 3.8. Substituting Q 2 into the demand equation to determine price: P = 50 – 5*3.8 = $31. The profits for Firm 2 will be: π 2 = (31)(3.8) - (10 + (12)(3.8)) = $62.20. b. What is each firm’s equilibrium output and profit if they behave noncooperatively? Use the Cournot model. Draw the firms’ reaction curves and show the equilibrium. In the Cournot model, Firm 1 takes Firm 2’s output as given and maximizes profits. The profit function derived in 2.a becomes π 1 = (50 - 5Q 1 - 5Q 2 )Q 1 - (20 + 10Q 1 ), or π = 40Q 1 − 5Q 1 2 − 5Q 1 Q 2 − 20. Setting the derivative of the profit function with respect to Q 1 to zero, we find Firm 1’s reaction function: ∂ π ∂ 1 Q = 40−10 1 Q-5 2 Q=0, or 1 Q=4- Q 2 2 ⎛ ⎝ ⎞ ⎠ . Similarly, Firm 2’s reaction function is Q 2 = 3.8− Q 1 2 ⎛ ⎝ ⎞ ⎠ . To find the Cournot equilibrium, we substitute Firm 2’s reaction function into Firm 1’s reaction function: Q 1 = 4 − 1 2 ⎛ ⎝ ⎞ ⎠ 3.8 − Q 1 2 ⎛ ⎝ ⎞ ⎠ , or Q 1 = 2.8. Substituting this value for Q 1 into the reaction function for Firm 2, we find Q 2 = 2.4. Substituting the values for Q 1 and Q 2 into the demand function to determine the equilibrium price: 192 Chapter 12: Monopolistic Competition and Oligopoly P = 50 – 5(2.8+2.4) = $24. The profits for Firms 1 and 2 are equal to π 1 = (24)(2.8) - (20 + (10)(2.8)) = 19.20 and π 2 = (24)(2.4) - (10 + (12)(2.4)) = 18.80. c. How much should Firm 1 be willing to pay to purchase Firm 2 if collusion is illegal but the takeover is not? In order to determine how much Firm 1 will be willing to pay to purchase Firm 2, we must compare Firm 1’s profits in the monopoly situation versus those in an oligopoly. The difference between the two will be what Firm 1 is willing to pay for Firm 2. From part a, profit of firm 1 when it set marginal revenue equal to its marginal cost was $60. This is what the firm would earn if it was a monopolist. From part b, profit was $19.20 for firm 1. Firm 1 would therefore be willing to pay up to $40.80 for firm 2. 3. A monopolist can produce at a constant average (and marginal) cost of AC = MC = 5. It faces a market demand curve given by Q = 53 - P. a. Calculate the profit-maximizing price and quantity for this monopolist. Also calculate its profits. The monopolist wants to choose quantity to maximize its profits: max π = PQ - C(Q), π = (53 - Q)(Q) - 5Q, or π = 48Q - Q 2 . To determine the profit-maximizing quantity, set the change in π with respect to the change in Q equal to zero and solve for Q: d dQ QQ π =−+= =2480 24,. or Substitute the profit-maximizing quantity, Q = 24, into the demand function to find price: 24 = 53 - P, or P = $29. Profits are equal to π = TR - TC = (29)(24) - (5)(24) = $576. b. Suppose a second firm enters the market. Let Q 1 be the output of the first firm and Q 2 be the output of the second. Market demand is now given by Q 1 + Q 2 = 53 - P. Assuming that this second firm has the same costs as the first, write the profits of each firm as functions of Q 1 and Q 2 . When the second firm enters, price can be written as a function of the output of two firms: P = 53 - Q 1 - Q 2 . We may write the profit functions for the two firms: π 1 = PQ 1 − CQ 1 ()= 53 −Q 1 − Q 2 ()Q 1 − 5Q 1 , or π 111 2 12 1 53 5=−−−QQQQ Q 193 Chapter 12: Monopolistic Competition and Oligopoly and π 2 = PQ 2 − CQ 2 ()= 53 − Q 1 − Q 2 ()Q 2 − 5Q 2 , or π 222 2 12 2 53 5=−−−QQQQ Q. c. Suppose (as in the Cournot model) that each firm chooses its profit-maximizing level of output on the assumption that its competitor’s output is fixed. Find each firm’s “reaction curve” (i.e., the rule that gives its desired output in terms of its competitor’s output). Under the Cournot assumption, Firm 1 treats the output of Firm 2 as a constant in its maximization of profits. Therefore, Firm 1 chooses Q 1 to maximize π 1 in b with Q 2 being treated as a constant. The change in π 1 with respect to a change in Q 1 is ∂ π ∂ 1 1 12 1 2 53 2 5 0 24 2Q QQ Q Q =− − −= =−,. or This equation is the reaction function for Firm 1, which generates the profit- maximizing level of output, given the constant output of Firm 2. Because the problem is symmetric, the reaction function for Firm 2 is Q Q 2 1 24 2 =−. d. Calculate the Cournot equilibrium (i.e., the values of Q 1 and Q 2 for which both firms are doing as well as they can given their competitors’ output). What are the resulting market price and profits of each firm? To find the level of output for each firm that would result in a stationary equilibrium, we solve for the values of Q 1 and Q 2 that satisfy both reaction functions by substituting the reaction function for Firm 2 into the one for Firm 1: Q 1 = 24 − 1 2 ⎛ ⎝ ⎞ ⎠ 24 − Q 1 2 ⎛ ⎝ ⎞ ⎠ , or Q 1 = 16. By symmetry, Q 2 = 16. To determine the price, substitute Q 1 and Q 2 into the demand equation: P = 53 - 16 - 16 = $21. Profits are given by π i = PQ i - C(Q i ) = π i = (21)(16) - (5)(16) = $256. Total profits in the industry are π 1 + π 2 = $256 +$256 = $512. *e. Suppose there are N firms in the industry, all with the same constant marginal cost, MC = 5. Find the Cournot equilibrium. How much will each firm produce, what will be the market price, and how much profit will each firm earn? Also, show that as N becomes large the market price approaches the price that would prevail under perfect competition. If there are N identical firms, then the price in the market will be P = 53 − Q 1 + Q 2 + L + Q N ( ) . 194 Chapter 12: Monopolistic Competition and Oligopoly Profits for the i’th firm are given by π i = PQ i − CQ i ( ) , π i = 53Q i − Q 1 Q i − Q 2 Q i − L − Q i 2 − L − Q N Q i − 5Q i . Differentiating to obtain the necessary first-order condition for profit maximization, d dQ QQQ i iN π = − −− −− −=53 2 5 0 1 LL . Solving for Q i , Q i = 24− 1 2 Q 1 +L + Q i−1 + Q i+1 +L + Q N () . If all firms face the same costs, they will all produce the same level of output, i.e., Q i = Q*. Therefore, Q* = 24− 1 2 N − 1()Q*, or 2Q* = 48− N −1()Q*, or N +1()Q* = 48, or Q* = 48 N + 1 () . We may substitute for Q = NQ*, total output, in the demand function: P = 53− N 48 N +1 ⎛ ⎝ ⎞ ⎠ . Total profits are π T = PQ - C(Q) = P(NQ*) - 5(NQ*) or π T = 53 − N 48 N + 1 ⎛ ⎝ ⎞ ⎠ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ N () 48 N + 1 ⎛ ⎝ ⎞ ⎠ − 5N 48 N +1 ⎛ ⎝ ⎞ ⎠ or π T = 48 − N () 48 N + 1 ⎛ ⎝ ⎞ ⎠ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ N () 48 N + 1 ⎛ ⎝ ⎞ ⎠ or π T = 48 () N + 1− N N + 1 ⎛ ⎝ ⎞ ⎠ 48 () N N +1 ⎛ ⎝ ⎞ ⎠ = 2, 304 () N N + 1 () 2 ⎛ ⎝ ⎞ ⎠ . Notice that with N firms Q = 48 N N + 1 ⎛ ⎝ ⎞ ⎠ and that, as N increases (N → ∞) Q = 48. Similarly, with P = 53 − 48 N N + 1 ⎛ ⎝ ⎞ ⎠ , as N → ∞, 195 Chapter 12: Monopolistic Competition and Oligopoly P = 53 - 48 = 5. With P = 5, Q = 53 - 5 = 48. Finally, π T = 2,304 N N +1 () 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ , so as N → ∞, π T = $0. In perfect competition, we know that profits are zero and price equals marginal cost. Here, π T = $0 and P = MC = 5. Thus, when N approaches infinity, this market approaches a perfectly competitive one. 4. This exercise is a continuation of Exercise 3. We return to two firms with the same constant average and marginal cost, AC = MC = 5, facing the market demand curve Q 1 + Q 2 = 53 - P. Now we will use the Stackelberg model to analyze what will happen if one of the firms makes its output decision before the other. a. Suppose Firm 1 is the Stackelberg leader (i.e., makes its output decisions before Firm 2). Find the reaction curves that tell each firm how much to produce in terms of the output of its competitor. Firm 1, the Stackelberg leader, will choose its output, Q 1 , to maximize its profits, subject to the reaction function of Firm 2: max π 1 = PQ 1 - C(Q 1 ), subject to Q 2 = 24 − Q 1 2 ⎛ ⎝ ⎞ ⎠ . Substitute for Q 2 in the demand function and, after solving for P, substitute for P in the profit function: max π 1 = 53− Q 1 − 24 − Q 1 2 ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ Q 1 ()− 5Q 1 . To determine the profit-maximizing quantity, we find the change in the profit function with respect to a change in Q 1 : d dQ QQ π 1 1 11 53 2 24 5=− −+−. Set this expression equal to 0 to determine the profit-maximizing quantity: 53 - 2Q 1 - 24 + Q 1 - 5 = 0, or Q 1 = 24. Substituting Q 1 = 24 into Firm 2’s reaction function gives Q 2 : Q 2 24 24 2 12=− =. Substitute Q 1 and Q 2 into the demand equation to find the price: 196 Chapter 12: Monopolistic Competition and Oligopoly P = 53 - 24 - 12 = $17. Profits for each firm are equal to total revenue minus total costs, or π 1 = (17)(24) - (5)(24) = $288 and π 2 = (17)(12) - (5)(12) = $144. Total industry profit, π T = π 1 + π 2 = $288 + $144 = $432. Compared to the Cournot equilibrium, total output has increased from 32 to 36, price has fallen from $21 to $17, and total profits have fallen from $512 to $432. Profits for Firm 1 have risen from $256 to $288, while the profits of Firm 2 have declined sharply from $256 to $144. b. How much will each firm produce, and what will its profit be? If each firm believes that it is the Stackelberg leader, while the other firm is the Cournot follower, they both will initially produce 24 units, so total output will be 48 units. The market price will be driven to $5, equal to marginal cost. It is impossible to specify exactly where the new equilibrium point will be, because no point is stable when both firms are trying to be the Stackelberg leader. 5. Two firms compete in selling identical widgets. They choose their output levels Q 1 and Q 2 simultaneously and face the demand curve P = 30 - Q, where Q = Q 1 + Q 2 . Until recently, both firms had zero marginal costs. Recent environmental regulations have increased Firm 2’s marginal cost to $15. Firm 1’s marginal cost remains constant at zero. True or false: As a result, the market price will rise to the monopoly level. True. If only one firm were in this market, it would charge a price of $15 a unit. Marginal revenue for this monopolist would be MR = 30 - 2Q, Profit maximization implies MR = MC, or 30 - 2Q = 0, Q = 15, (using the demand curve) P = 15. The current situation is a Cournot game where Firm 1's marginal costs are zero and Firm 2's marginal costs are 15. We need to find the best response functions: Firm 1’s revenue is PQ 1 = (30 − Q 1 − Q 2 )Q 1 = 30Q 1 − Q 1 2 − Q 1 Q 2 , and its marginal revenue is given by: M R 1 = 30 − 2Q 1 − Q 2 . Profit maximization implies MR 1 = MC 1 or 30 − 2Q 1 − Q 2 = 0 ⇒ Q 1 =15− Q 2 2 , which is Firm 1’s best response function. 197 Chapter 12: Monopolistic Competition and Oligopoly Firm 2’s revenue function is symmetric to that of Firm 1 and hence M R 2 = 30 − Q 1 − 2Q 2 . Profit maximization implies MR 2 = MC 2 , or 30 − 2Q 2 − Q 1 = 15⇒ Q 2 = 7.5 − Q 1 2 , which is Firm 2’s best response function. Cournot equilibrium occurs at the intersection of best response functions. Substituting for Q 1 in the response function for Firm 2 yields: Q 2 = 7.5− 0.5(15 − Q 2 2 ). Thus Q 2 =0 and Q 1 =15. P = 30 - Q 1 + Q 2 = 15, which is the monopoly price. 6. Suppose that two identical firms produce widgets and that they are the only firms in the market. Their costs are given by C 1 = 60Q 1 and C 2 = 60Q 2 , where Q 1 is the output of Firm 1 and Q 2 the output of Firm 2. Price is determined by the following demand curve: P = 300 - Q where Q = Q 1 + Q 2 . a. Find the Cournot-Nash equilibrium. Calculate the profit of each firm at this equilibrium. To determine the Cournot-Nash equilibrium, we first calculate the reaction function for each firm, then solve for price, quantity, and profit. Profit for Firm 1, TR 1 - TC 1 , is equal to π 1 = 300Q 1 − Q 1 2 − Q 1 Q 2 − 60Q 1 = 240Q 1 − Q 1 2 − Q 1 Q 2 . Therefore, ∂ 1 π ∂ 1 Q = 240 − 2 1 Q − 2 Q. Setting this equal to zero and solving for Q 1 in terms of Q 2 : Q 1 = 120 - 0.5Q 2 . This is Firm 1’s reaction function. Because Firm 2 has the same cost structure, Firm 2’s reaction function is Q 2 = 120 - 0.5Q 1 . Substituting for Q 2 in the reaction function for Firm 1, and solving for Q 1 , we find Q 1 = 120 - (0.5)(120 - 0.5Q 1 ), or Q 1 = 80. By symmetry, Q 2 = 80. Substituting Q 1 and Q 2 into the demand equation to determine the price at profit maximization: P = 300 - 80 - 80 = $140. Substituting the values for price and quantity into the profit function, π 1 = (140)(80) - (60)(80) = $6,400 and 198 Chapter 12: Monopolistic Competition and Oligopoly π 2 = (140)(80) - (60)(80) = $6,400. Therefore, profit is $6,400 for both firms in Cournot-Nash equilibrium. b. Suppose the two firms form a cartel to maximize joint profits. How many widgets will be produced? Calculate each firm’s profit. Given the demand curve is P=300-Q, the marginal revenue curve is MR=300-2Q. Profit will be maximized by finding the level of output such that marginal revenue is equal to marginal cost: 300-2Q=60 Q=120. When output is equal to 120, price will be equal to 180, based on the demand curve. Since both firms have the same marginal cost, they will split the total output evenly between themselves so they each produce 60 units. Profit for each firm is: π = 180(60)-60(60)=$7,200. Note that the other way to solve this problem, and arrive at the same solution is to use the profit function for either firm from part a above and let Q = Q 1 = Q 2 . c. Suppose Firm 1 were the only firm in the industry. How would the market output and Firm 1’s profit differ from that found in part (b) above? If Firm 1 were the only firm, it would produce where marginal revenue is equal to marginal cost, as found in part b. In this case firm 1 would produce the entire 120 units of output and earn a profit of $14,400. d. Returning to the duopoly of part (b), suppose Firm 1 abides by the agreement, but Firm 2 cheats by increasing production. How many widgets will Firm 2 produce? What will be each firm’s profits? Assuming their agreement is to split the market equally, Firm 1 produces 60 widgets. Firm 2 cheats by producing its profit-maximizing level, given Q 1 = 60. Substituting Q 1 = 60 into Firm 2’s reaction function: Q 2 = 120− 60 2 = 90. Total industry output, Q T , is equal to Q 1 plus Q 2 : Q T = 60 + 90 = 150. Substituting Q T into the demand equation to determine price: P = 300 - 150 = $150. 199 Chapter 12: Monopolistic Competition and Oligopoly 200 Substituting Q 1 , Q 2 , and P into the profit function: π 1 = (150)(60) - (60)(60) = $5,400 and π 2 = (150)(90) - (60)(90) = $8,100. Firm 2 has increased its profits at the expense of Firm 1 by cheating on the agreement. 7. Suppose that two competing firms, A and B, produce a homogeneous good. Both firms have a marginal cost of MC=$50. Describe what would happen to output and price in each of the following situations if the firms are at (i) Cournot equilibrium, (ii) collusive equilibrium, and (iii) Bertrand equilibrium. a. Firm A must increase wages and its MC increases to $80. (i) In a Cournot equilibrium you must think about the effect on the reaction functions, as illustrated in figure 12.4 of the text. When firm A experiences an increase in marginal cost, their reaction function will shift inwards. The quantity produced by firm A will decrease and the quantity produced by firm B will increase. Total quantity produced will tend to decrease and price will increase. (ii) In a collusive equilibrium, the two firms will collectively act like a monopolist. When the marginal cost of firm A increases, firm A will reduce their production. This will increase price and cause firm B to increase production. Price will be higher and total quantity produced will be lower. (iii) Given that the good is homogeneous, both will produce where price equals marginal cost. Firm A will increase price to $80 and firm B will keep its price at $50. Assuming firm B can produce enough output, they will supply the entire market. b. The marginal cost of both firms increases. (i) Again refer to figure 12.4. The increase in the marginal cost of both firms will shift both reaction functions inwards. Both firms will decrease quantity produced and price will increase. (ii) When marginal cost increases, both firms will produce less and price will increase, as in the monopoly case. (iii) As in the above cases, price will increase and quantity produced will decrease. c. The demand curve shifts to the right. (i) This is the opposite of the above case in part b. In this case, both reaction functions will shift outwards and both will produce a higher quantity. Price will tend to increase. (ii) Both firms will increase the quantity produced as demand and marginal revenue increase. Price will also tend to increase. (iii) Both firms will supply more output. Given that marginal cost is constant, the price will not change. 8. Suppose the airline industry consisted of only two firms: American and Texas Air Corp. Let the two firms have identical cost functions, C(q) = 40q. Assume the demand curve for [...]... non-OPEC * production is Q Non −OPEC Notice that the curves must be drawn accurately to give this result, and again have been drawn in a linear fashion as opposed to non-linear for ease of accuracy Price S* S P** P* D* = W* - S* DW D=W-S Q* Non-OPEC Q Non-OPEC MR Q OPEC QW MR * Q*D Figure 12. 12.a.ii 212 Quantity Chapter 12: Monopolistic Competition and Oligopoly b Calculate OPEC’s optimal (profit-maximizing)... 300 - 6q1 - 3q2 = 30 + 3q1 q1 = 30 - (1/3)q2 By symmetry, BBBS’s best response function will be: 206 Chapter 12: Monopolistic Competition and Oligopoly q2 = 30 - (1/3)q1 Cournot equilibrium occurs at the intersection of these two best response functions, given by: q1 = q2 = 22.5 Thus, Q = q1 + q2 = 45 P = 300 - 3(45) = $165 Profit for both firms will be equal and given by: 2 R - C = (165) (22.5) - (30(22.5)... (55)(15) - (40)(15) = $225 Second, with investment by both firms, the reaction functions would be: Q1 = 37.5 - 0.5Q2 and Q2 = 37.5 - 0.5Q1 To determine Q1, substitute for Q2 in the first reaction function and solve for Q1: Q1 = 37.5 - (0.5)(37.5 - 0.5Q1) = 25 Substituting for Q1 in the second reaction function to find Q2: 202 Chapter 12: Monopolistic Competition and Oligopoly Q2 = 37.5 - 0.5(37.5 - 0.5Q2)... production can be read off of the non-OPEC supply curve at a price of P* Note that in the two figures below, the demand and supply curves are actually nonlinear They have been drawn in a linear fashion for ease of accuracy 211 Chapter 12: Monopolistic Competition and Oligopoly Price S P* DW D=W-S QNon-OPEC MR QOPEC QW Quantity Figure 12. 12.a.i Next, suppose non-OPEC oil becomes more expensive Then... )q1 − (30q1 + 1.5q12 ) Π = 270q1 − 4.5q12 − 3q1q2 1 Π = 270q1 − 4.5q12 − 3q1 (30 − q1 ) 3 2 Π = 180q1 − 3.5q1 Profit maximization implies: ∂Π = 180 − 7q1 = 0 ∂q1 This results in q1=25.7 and q2=21.4 The equilibrium price and profits will then be: P = 200 - 2(q1 + q2) = 200 - 2(25.7 + 21.4) = $158.57 2 π1 = (158.57) (25.7) - (30) (25.7) – 1.5*25.7 = $2313.51 2 π2 = (158.57) (21.4) - (30) (21.4) – 1.5*21.4... the profit-maximizing price, find the change in profit with respect to a change in price: 209 Chapter 12: Monopolistic Competition and Oligopoly dπ 1 = 30 − P1 dP 1 Set this expression equal to zero to find the profit-maximizing price: 30 - P1 = 0, or P1 = $30 Substitute P1 in Firm 2’s reaction function to find P2: P2 = 20 + 30 = $25 2 At these prices, Q1 = 20 - 30 + 25 = 15 and Q2 = 20 + 30 - 25 =... determine price: P = 100 - 50 = $50 Therefore, American’s profits if Q1 = Q2 = 25 (when both firms have MC = AC = 25) are π2 = (100 - 25 - 25)(25) - (25)(25) = $625 The difference in profit with and without the cost-saving investment for American is $400 American would be willing to invest up to $400 to reduce its marginal cost to 25 if Texas Air also has marginal costs of 25 203 Chapter 12: Monopolistic Competition... function P = 300 - 3Q where total output Q is the sum of each firm’s output q1 and q2 We find the best response functions for both firms by setting marginal revenue equal to marginal cost (alternatively you can set up the profit function for each firm and differentiate with respect to the quantity produced for that firm): 2 R1 = P q1 = (300 - 3(q1 + q2)) q1 = 300q1 - 3q1 - 3q1q2 MR1 = 300 - 6q1 - 3q2 MC1... Q2 in the reaction function for Texas Air, Q1 = 30 - 0.5(30 - 0.5Q1) = 20 By symmetry, Q2 = 20 Industry output, QT, is Q1 plus Q2, or QT = 20 + 20 = 40 Substituting industry output into the demand equation, we find P = 60 Substituting Q1, Q2, and P into the profit function, we find 2 π1 = π2 = 60(20) -2 0 - (20)(20) = $400 for both firms in Cournot-Nash equilibrium b What would be the equilibrium quantity... Joint profits will be (30 0-3 Q)Q - 2(30(Q/2) + 1.5(Q/2) ) = 270Q - 3.75Q and will be maximized at Q = 36 You can find this quantity by differentiating the above profit function with respect to Q, setting the resulting first order condition equal to zero, and then solving for Q Thus, we will have q1 = q2 = 36 / 2 = 18 and P = 300 - 3(36) = $192 2 Profit for each firm will be 18(192) - (30(18) + 1.5(18 )) . (300 - 3(q 1 + q 2 )) q 1 = 300q 1 - 3q 1 2 - 3q 1 q 2 . MR 1 = 300 - 6q 1 - 3q 2 MC 1 = 30 + 3q 1 300 - 6q 1 - 3q 2 = 30 + 3q 1 q 1 = 30 -. quantity to maximize its profits: max π = PQ - C(Q), π = (53 - Q)(Q) - 5Q, or π = 48Q - Q 2 . To determine the profit-maximizing quantity, set the change in